International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 13, Number 6 (2020), pp. 1147-1158 © International Research Publication House. http://www.irphouse.com 1147 Elastic Buckling Analysis of SSCF and SSSS Rectangular Thin Plates using the Single Finite Fourier Sine Integral Transform Method Michael Ebie Onyia 1 , Eghosa. O. Rowland-Lato 2 and Charles Chinwuba Ike 3 1 Department of Civil Engineering University of Nigeria, Nsukka, Enugu State, Nigeria. [email protected], ORCID: 0000-0002-0956-0077 2 Department of Civil Engineering, University of Port Harcourt, Choba. Nigeria. [email protected], ORCID: 0000-0001-7289-0530 3 Department of Civil Engineering, Enugu State University of Science and Technology, Enugu. [email protected], ORCID: 0000-0001-6952-0993 Abstract The analysis of the stability problem of plate subjected to in- plane compressive load is important due to the relatively poor capacity of plates in resisting compressive forces compared to tensile forces. It is also significant due to the nonlinear, sudden nature of buckling failures. This study presents the elastic buckling analysis of SSCF and SSSS rectangular thin plates using the single finite Fourier sine integral transform method. The considered plate problems are (i) rectangular thin plate simply supported on two opposite edges, clamped on one edge and free on the fourth edge; (ii) rectangular thin plate simply supported on all edges. The plates are subject to uniaxial uniform compressive loads on the two simply supported edges. The governing domain equation is a fourth order partial differential equation (PDE). The problem solved is a boundary value problem (BVP) since the domain PDE is subject to the boundary conditions at the four edges. The single finite sine transform adopted automatically satisfies the Dirichlet boundary conditions along the simply supported edges. The transform converts the BVP to an integral equation, which simplifies upon use of the linearity properties and integration by parts to a system of homogeneous ordinary differential equations (ODEs) in terms of the transform of the unknown buckling deflection. The general solution of the system of ODEs is obtained using trial function methods. Enforcement of boundary conditions along the y = 0, and y = b edges (for the SSCF and SSSS plates considered) results in a system of four sets of homogeneous equations in terms of the integration constants. The characteristic buckling equation in each case considered is found for nontrivial solutions as a transcendental equation, whose roots are used to obtain the buckling loads for various values of the aspect ratio and for any buckling modes. In each considered case, the obtained buckling equation is exact and identical with exact expressions previously obtained in the literature using other solution methods. The buckling loads obtained by the present method are validated by the observed agreement with results obtained by previous researchers who used other methods. Keywords: single finite Fourier sine integral transform method, characteristic elastic buckling equation, critical elastic buckling load, elastic buckling load coefficient, elastic buckling problem. I. INTRODUCTION The analysis of stability problems of plate carrying in-plane compressive loads is important in structures due to the relatively poor capacity of plates in resisting compressive forces [1 – 7]. It is also important due to the nonlinear, sudden nature of buckling. A good knowledge of buckling loads and the associated buckling mode shapes is fundamental to structural analysis and design for compressed plates; hence the need for development of effective analytical and numerical methods for solving the buckling problem. The determination of buckling loads is thus an important theme in structural analysis. In-plane compressive loads can be uniformly or non-uniformly applied over the edges uniaxially or biaxially. The plate may be modelled as a thin or thick plate, depending on the thickness-width ratio, where the width is the least in-plane dimension. The buckling problem is classified as elastic or inelastic. In elastic buckling problems, the critical buckling load is smaller than the elastic limit of the material otherwise the problem is called inelastic buckling. Navier’s investigation of stability problem of rectangular thin plates was one of the first studies on the subject of thin plate stability. Navier used the assumptions and hypotheses of the Kirchhoff thin plate theory to derive the stability equation of rectangular thin plates that included the twisting term. Saint-Venant later presented a modification of the Navier’s equation to include the axially applied edge loads and the shearing loads. Saint-Venant’s modified differential equation provided the theoretical basis for the elastic stability of thin plates with various edge loads and edge support conditions. Bryan used the total potential energy minimization principle to solve the elastic stability of a rectangular thin plate under uniaxial compressive load for Dirichlet boundary conditions. Bryan assumed the buckling shape function for the problem as a double Fourier sine series. Timoshenko [6] also presented solutions to elastic stability problem of rectangular thin plates with simply supported edges under uniaxial compressive loads by assuming the buckling shape function as series of sinusoidal half waves in the
Elastic Buckling Analysis of SSCF and SSSS Rectangular Thin Plates using the Single Finite Fourier Sine Integral Transform Method
May 20, 2023
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